On the Learning Dynamics of Attention Networks

Attention models are typically learned by optimizing one of three standard loss functions that are variously called -- soft attention, hard attention, and latent variable marginal likelihood (LVML) attention. All three paradigms are motivated by the same goal of finding two models -- a `focus' model that `selects' the right \textit{segment} of the input and a `classification' model that processes the selected segment into the target label. However, they differ significantly in the way the selected segments are aggregated, resulting in distinct dynamics and final results. We observe a unique signature of models learned using these paradigms and explain this as a consequence of the evolution of the classification model under gradient descent when the focus model is fixed. We also analyze these paradigms in a simple setting and derive closed-form expressions for the parameter trajectory under gradient flow. With the soft attention loss, the focus model improves quickly at initialization and splutters later on. On the other hand, hard attention loss behaves in the opposite fashion. Based on our observations, we propose a simple hybrid approach that combines the advantages of the different loss functions and demonstrates it on a collection of semi-synthetic and real-world datasets

On the Interpretability of Attention Networks

Attention mechanisms form a core component of several successful deep learning architectures, and are based on one key idea: ''The output depends only on a small (but unknown) segment of the input.'' In several practical applications like image captioning and language translation, this is mostly true. In trained models with an attention mechanism, the outputs of an intermediate module that encodes the segment of input responsible for the output is often used as a way to peek into the `reasoning` of the network. We make such a notion more precise for a variant of the classification problem that we term selective dependence classification (SDC) when used with attention model architectures. Under such a setting, we demonstrate various error modes where an attention model can be accurate but fail to be interpretable, and show that such models do occur as a result of training. We illustrate various situations that can accentuate and mitigate this behaviour. Finally, we use our objective definition of interpretability for SDC tasks to evaluate a few attention model learning algorithms designed to encourage sparsity and demonstrate that these algorithms help improve interpretability.

Consistent Multiclass Algorithms for Complex Metrics and Constraints

We present consistent algorithms for multiclass learning with complex performance metrics and constraints, where the objective and constraints are defined by arbitrary functions of the confusion matrix. This setting includes many common performance metrics such as the multiclass G-mean and micro F1-measure, and constraints such as those on the classifier's precision and recall and more recent measures of fairness discrepancy. We give a general framework for designing consistent algorithms for such complex design goals by viewing the learning problem as an optimization problem over the set of feasible confusion matrices. We provide multiple instantiations of our framework under different assumptions on the performance metrics and constraints, and in each case show rates of convergence to the optimal (feasible) classifier (and thus asymptotic consistency). Experiments on a variety of multiclass classification tasks and fairness-constrained problems show that our algorithms compare favorably to the state-of-the-art baselines.

Predicting the success of Gradient Descent for a particular Dataset-Architecture-Initialization (DAI)

Despite their massive success, training successful deep neural networks still largely relies on experimentally choosing an architecture, hyper-parameters, initialization, and training mechanism. In this work, we focus on determining the success of standard gradient descent method for training deep neural networks on a specified dataset, architecture, and initialization (DAI) combination. Through extensive systematic experiments, we show that the evolution of singular values of the matrix obtained from the hidden layers of a DNN can aid in determining the success of gradient descent technique to train a DAI, even in the absence of validation labels in the supervised learning paradigm. This phenomenon can facilitate early give-up, stopping the training of neural networks which are predicted to not generalize well, early in the training process. Our experimentation across multiple datasets, architectures, and initializations reveals that the proposed scores can more accurately predict the success of a DAI than simply relying on the validation accuracy at earlier epochs to make a judgment.

Using noise resilience for ranking generalization of deep neural networks

Recent papers have shown that sufficiently overparameterized neural networks can perfectly fit even random labels. Thus, it is crucial to understand the underlying reason behind the generalization performance of a network on real-world data. In this work, we propose several measures to predict the generalization error of a network given the training data and its parameters. Using one of these measures, based on noise resilience of the network, we secured 5th position in the predicting generalization in deep learning (PGDL) competition at NeurIPS 2020.

Inductive Bias of Gradient Descent for Exponentially Weight Normalized Smooth Homogeneous Neural Nets

We analyze the inductive bias of gradient descent for weight normalized smooth homogeneous neural nets, when trained on exponential or cross-entropy loss. Our analysis focuses on exponential weight normalization (EWN), which encourages weight updates along the radial direction. This paper shows that the gradient flow path with EWN is equivalent to gradient flow on standard networks with an adaptive learning rate, and hence causes the weights to be updated in a way that prefers asymptotic relative sparsity. These results can be extended to hold for gradient descent via an appropriate adaptive learning rate. The asymptotic convergence rate of the loss in this setting is given by $\Theta(\frac{1}{t(\log t)^2})$, and is independent of the depth of the network. We contrast these results with the inductive bias of standard weight normalization (SWN) and unnormalized architectures, and demonstrate their implications on synthetic data sets.Experimental results on simple data sets and architectures support our claim on sparse EWN solutions, even with SGD. This demonstrates its potential applications in learning prunable neural networks.

Convex Calibrated Surrogates for the Multi-Label F-Measure

The F-measure is a widely used performance measure for multi-label classification, where multiple labels can be active in an instance simultaneously (e.g. in image tagging, multiple tags can be active in any image). In particular, the F-measure explicitly balances recall (fraction of active labels predicted to be active) and precision (fraction of labels predicted to be active that are actually so), both of which are important in evaluating the overall performance of a multi-label classifier. As with most discrete prediction problems, however, directly optimizing the F-measure is computationally hard. In this paper, we explore the question of designing convex surrogate losses that are calibrated for the F-measure -- specifically, that have the property that minimizing the surrogate loss yields (in the limit of sufficient data) a Bayes optimal multi-label classifier for the F-measure. We show that the F-measure for an $s$-label problem, when viewed as a $2^s \times 2^s$ loss matrix, has rank at most $s^2+1$, and apply a result of Ramaswamy et al. (2014) to design a family of convex calibrated surrogates for the F-measure. The resulting surrogate risk minimization algorithms can be viewed as decomposing the multi-label F-measure learning problem into $s^2+1$ binary class probability estimation problems. We also provide a quantitative regret transfer bound for our surrogates, which allows any regret guarantees for the binary problems to be transferred to regret guarantees for the overall F-measure problem, and discuss a connection with the algorithm of Dembczynski et al. (2013). Our experiments confirm our theoretical findings.

On Controllable Sparse Alternatives to Softmax

Converting an n-dimensional vector to a probability distribution over n objects is a commonly used component in many machine learning tasks like multiclass classification, multilabel classification, attention mechanisms etc. For this, several probability mapping functions have been proposed and employed in literature such as softmax, sum-normalization, spherical softmax, and sparsemax, but there is very little understanding in terms how they relate with each other. Further, none of the above formulations offer an explicit control over the degree of sparsity. To address this, we develop a unified framework that encompasses all these formulations as special cases. This framework ensures simple closed-form solutions and existence of sub-gradients suitable for learning via backpropagation. Within this framework, we propose two novel sparse formulations, sparsegen-lin and sparsehourglass, that seek to provide a control over the degree of desired sparsity. We further develop novel convex loss functions that help induce the behavior of aforementioned formulations in the multilabel classification setting, showing improved performance. We also demonstrate empirically that the proposed formulations, when used to compute attention weights, achieve better or comparable performance on standard seq2seq tasks like neural machine translation and abstractive summarization.

Mixture Proportion Estimation via Kernel Embedding of Distributions

Mixture proportion estimation (MPE) is the problem of estimating the weight of a component distribution in a mixture, given samples from the mixture and component. This problem constitutes a key part in many "weakly supervised learning" problems like learning with positive and unlabelled samples, learning with label noise, anomaly detection and crowdsourcing. While there have been several methods proposed to solve this problem, to the best of our knowledge no efficient algorithm with a proven convergence rate towards the true proportion exists for this problem. We fill this gap by constructing a provably correct algorithm for MPE, and derive convergence rates under certain assumptions on the distribution. Our method is based on embedding distributions onto an RKHS, and implementing it only requires solving a simple convex quadratic programming problem a few times. We run our algorithm on several standard classification datasets, and demonstrate that it performs comparably to or better than other algorithms on most datasets.

Convex Calibration Dimension for Multiclass Loss Matrices

We study consistency properties of surrogate loss functions for general multiclass learning problems, defined by a general multiclass loss matrix. We extend the notion of classification calibration, which has been studied for binary and multiclass 0-1 classification problems (and for certain other specific learning problems), to the general multiclass setting, and derive necessary and sufficient conditions for a surrogate loss to be calibrated with respect to a loss matrix in this setting. We then introduce the notion of convex calibration dimension of a multiclass loss matrix, which measures the smallest `size' of a prediction space in which it is possible to design a convex surrogate that is calibrated with respect to the loss matrix. We derive both upper and lower bounds on this quantity, and use these results to analyze various loss matrices. In particular, we apply our framework to study various subset ranking losses, and use the convex calibration dimension as a tool to show both the existence and non-existence of various types of convex calibrated surrogates for these losses. Our results strengthen recent results of Duchi et al. (2010) and Calauzenes et al. (2012) on the non-existence of certain types of convex calibrated surrogates in subset ranking. We anticipate the convex calibration dimension may prove to be a useful tool in the study and design of surrogate losses for general multiclass learning problems.